NavList:

I am pretty sure that we are not using the same terminology as to what a "parameter" is, hence the miscommunication.

LOPs are indeed lines, and hence are one-dimensional objects, therefore needing one parameter.  Their "sizes" are measured in units of length.

The Earth's surface is a two-dimensional object, parametrized by latitude and longitude.  Its "size" is measured in units of area, which is length squared.

Aircraft operate in the atmosphere; there we need three parameters, such as latitude, longitude, and altitude.  For submarines we have depth as the third parameter.  The "sizes" of the atmosphere and the ocean are measured in the units of volume = length cubed.

Here are three (admittedly trivial) examples to illustrate my point.  For the general case I'd have to work out the equations for which I don't have the time right now.   I believe, however, that NavList member Andres Ruiz, whose excellent software does in fact plot circles of equal altitude, must have already worked this out in full generality.  I invite Andres to comment.

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Example 1: Polaris exactly above the North Pole, Ho = 40.
Solution: The LOP is the parallel 40 degrees North.
Parametrization: Parameter P covers the interval (-180,180], i.e. the 360 degree-span of a full circle:

Latitude = Ho = 40
Longitude = P
Note that the Ho is not a parameter, it is a constant related to the size of the circle.  Constants like Ho are fixed for a given LOP, while the parameter P runs freely across its interval, thus enumerating all the points on the LOP.

There is an infinite number of possible parametrizations; another one for this example has P covering the interval [0,360):
Latitude = Ho
Longitude = - P, for 0 <= P < 180
                   = 360 - P, for 180 <= P < 360
In this case the parameter P is the GHA.

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Example 2: GP given by GHA = 10 and Dec = 0, Ho = 0, hence ZD = 90.
Solution: The LOP consists of the meridians 80 degrees East and 100 degrees West.
Parametrization: Parameter P covers the interval [0,360), the full circle again:

For 0 <= P <= 180
Latitude = 90 - P
Longitude = - ( GHA + ZD ) = -100

For 180 < P < 360
Latitude = P - 270
Longitude = - ( GHA - ZD ) = +80

The GHA, Dec, and Ho (ZD) are constants that uniquely define the LOP.  In this example the parameter P is the distance/(60 nm per degree) covered to get to the point P on a circumnavigation trip starting at the North Pole, and initially heading in the GHA = 100 direction.

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Example 3: GP given by GHA and Dec, Ho very near 90, hence we have a non-zero but small ZD.
Solution: The LOP is a circle centered at the GP with radius ZD.  This LOP can be approximately computed using planar geometry.
Parametrization: Parameter P covers the interval [0,360):

Latitude = Dec  +  ZD * cos(P)
Longitude = Lon  +  ZD * sin(P) / cos(Dec)

where Lon is derived from GHA in the usual manner.  The parameter P is the azimuth in this case.

If GP is one of the Poles, then use the solution of Example 1 instead of this.

Now if we allow the Ho to become a free parameter also (and remove the planar restriction in the above equations), then instead of an LOP we could parametrize the entire Earth's surface with two parameters Ho and P.  Our usual grid of parallels and meridians is in fact a special case of such a parametrization, with GP -> North Pole, Ho -> latitude, and P -> longitude.

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Expressions in these examples are special cases of more complicated general formulae that include three constants GHA, Dec, Ho, and one parameter P in both the Latitude and Longitude equations.  Thus the complete LOP can be calculated "directly" without any AP.  At least, that is my understanding, perhaps John can comment on this as well.


Peter Hakel




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